with rational 0 < < 1. But all the angles of integer triangles must have rational cosines and this will occur only when i.e. the integer triangle is equilateral. The square of each internal angle bisector of an integer triangle is rational, because the general triangle formula for the internal angle bisector of angle A is where s is the semiperimeter (and likewise for the other angles' bisectors). Side split by an altitude Any altitude dropped from a vertex onto an opposite side or its extension will split that side or its extension into rational lengths. Medians The square of twice any median of an integer triangle is an integer, because the general formula for the squared median ma2 to side a is , giving (2ma)2 = 2b2 + 2c2 − a2 (and likewise for the medians to the other sides). Circumradius and inradius Because the square of the area of an integer triangle is rational, the square of its circumradius is also rational, as is the square of the inradius. The ratio of the inradius to the circumradius of an integer triangle is rational, equaling for semiperimeter s and area T. The product of the inradius and the circumradius of an integer triangle is rational, equaling Thus the squared distance between the incenter and the circumcenter of an integer triangle, given by Euler's theorem as R2 − 2Rr, is rational. Heronian triangles All Heronian triangles can be placed on a lattice with each vertex at a lattice point. General formula A Heronian triangle, also known as a Heron triangle or a Hero triangle, is a triangle with integer sides and integer area. Every Heronian triangle has sides proportional to for integers m, n and k subject to the constraints: The proportionality factor is generally a rational where q = gcd(a,b,c) reduces the generated Heronian triangle to its primitive and scales up this primitive to the required size. Pythagorean triangles A Pythagorean triangle is right-angled and Heronian. Its three integer sides are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad. All Pythagorean triples with hypotenuse which are primitive (the sides having no common factor) can be generated by where m and n are coprime integers and one of them is even with m > n. Every even number greater than 2 can be the leg of a Pythagorean triangle (not necessarily primitive) because if the leg is